Question

Problem 2. Solve the equations which are either exact or can be changed to the exact equations a. sint cos edt sin cdr 0 b.y = y(1) = 2

Answer 1

2. The given differential equation is sint cosxdt - sina dx=0 Here M = -sinx N- Sint cos x am ot - 0 ON aze -Sint sina Hence the equation is not exact. Now, (2 - ) 1 N sint sina sint cosa sina cosa f(x) and hence s f(x) dx e s sina cosa =e da - e - log (cosa) log(cosx)-1 e - (cosa)- is cos an I.F.

Multiplying by I.F, the equation becomes sina Cosa da + sintat = ora sinae dat COSX + S sint dt-C sint dt = c [c = I.C] log (eosa) - cost which is the required general solution. or, b. The given differential equation is y's ху 1 - x2 on, dy da axy 2-22 on, axy da (2-22 ) dy da +(22-2) dy = 0 or, Here M=2ay N = x2-2 2xy OM 29 -22 IN ox -2x ОМ) by exact. ON ox & hence the equation is

1. SM dx = S 2xy dx (as if y is constant) - 2. SN dy = S (322- 2) dy (as if se is constant) x²y-zy 3. Us Dy x²y-ay 4. The required Primitive is x² y - 2y = =C (e = arbitrar У constant ) Now y(1)=2 gives 2-4 = @ => C = -2 Therefore, the gener al Solution y (32-2) +2=0